Article

# Sur la capitulation des 2-classes d’idéaux de Q(\sqrt{d} , i).

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## Abstract

Let d ε N, i = √−1, k = Q (√d, i), k1 be the 2-Hilbert class field of k, k2 the 2-Hilbert class field of k1, Ck, 2 the 2-component of the class group of k and G2 the Galois group of k2/k. We suppose that the group Ck, 2 is of type (2, 2); then k1 contains three extensions Fi/k, i = 1, 2, 3. The aim of this Note is to study the problem of capitulation of the 2-ideal classes of k in Fi, i = 1, 2, 3, and to determine the structure of G2.

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... For a quadratic ield k, using Gauss's genus theory, one can easily deduce the rank of Cl 2 (k) . For biquadratic ields, the authors of [1,3] determined all positive integers d such that Cl 2 (k) is isomorphic to ℤ∕2ℤ × ℤ∕2ℤ , where k = ℚ( √ d, √ − ) and = 1, 2 . The papers [4,5] investigated the rank of Cl 2 (k) for the ields k = ℚ( √ d, √ m) , where m is a prime and d a positive square-free integer. ...
... The papers [4,5] investigated the rank of Cl 2 (k) for the ields k = ℚ( √ d, √ m) , where m is a prime and d a positive square-free integer. In the same direction, [7] classiied all ields k = ℚ( √ d, i) such that Cl 2 (k) is isomorphic to ℤ∕2ℤ × ℤ∕4ℤ or (ℤ∕2ℤ) 3 . In [9,11], E. Brown and C. Parry determined some imaginary quartic cyclic number ields k such that the rank of Cl 2 (k) is at most 3. Finally, [24] determined all imaginary biquadratic ields whose 2-class group is cyclic. ...
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Let d be an odd square-free integer and $$\zeta _8$$ a primitive 8-th root of unity. The purpose of this paper is to investigate the rank of the 2-class group of the fields $$L_d={\mathbb {Q}}(\zeta _8,\sqrt{d})$$.
... Editor R. Kučera. i ∈ {1, 2, 3}, à la structure du groupe G. Dans [1], A. Azizi a étudié le problème de capitulation des 2-classes d'idéaux des corps K = Q( √ d, i) où d est un entier naturel sans facteurs carrés et C 2,K est de type (2, 2), en utilisant les unités, il a trouvé le nombre des classes de K qui capitulent dans les sous-extensions propres de K ...
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Let $K=k\big (\sqrt{-p{\varepsilon }\sqrt{l}}\big )$ with $k={\mathbb{Q}}(\sqrt{l})$ where $l$ is a prime number such that $l=2$ or $l\equiv 5\;\@mod \;8$, $\varepsilon$ the fundamental unit of $k$, $p$ a prime number such that $p\equiv 1\;\@mod \;4$ and ${(\frac{p}{l})}_4=-1$, $K_2^{(1)}$ the Hilbert $2$-class field of $K$, $K_2^{(2)}$ the Hilbert $2$-class field of $K_2^{(1)}$ and $G=\operatorname{Gal\,}(K_2^{(2)}/K)$ the Galois group of $K_2^{(2)}/K$. According to E. Brown and C. J. Parry [7] and [8], $C_{2,K}$, the Sylow $2$-subgroup of the ideal class group of $K$, is isomorphic to ${\mathbb{Z}}/2{\mathbb{Z}}\times {\mathbb{Z}}/{2\mathbb{Z}}$, consequently $K_2^{(1)}/K$ contains three extensions $F_i/K$ $(i=1,2,3)$ and the tower of the Hilbert $2$-class field of $K$ terminates at either $K_2^{(1)}$ or $K_2^{(2)}$. In this work, we are interested in the problem of capitulation of the classes of $C_{2,K}$ in $F_i$ $(i=1,2,3)$ and to determine the structure of $G$.
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Let q1 and q2 be two prime integers such that q1≡7(mod8) and q2≡3(mod8). In this work we determine the unit groups of the fields L=Q(√q1,√q2,√2,i) and L+=Q(√q1,√q2,√2). Furthermore, as applications we compute the Iwasawa λ-invariant for some multiquadratic number fields of the form F=Q(√q1,√q2,i).
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Let p be a prime integer and L=Q(ζ8,p). In this paper, we determine the fields L for which the 2-class group is of type (2, 8) and those for which the 16-rank of the 2-class group equals 1.
Chapter
In this paper, we will determine the Pólya group of the field $$k = \mathbb{Q}(\sqrt{d},i)$$ and then we deduce a new method to characterize biquadratic Pólya fields $$k = \mathbb{Q}(\sqrt{d},i)$$. The capitulation theory allows us to construct a family of imaginary triquadratic Pólya fields.
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On the rank of the 2 2 -class group of Q ( m , d ) Q({\sqrt {m}},{\sqrt {d}}) . Let d d be a square-free positive integer and p p be a prime such that p ≡ 1 ( m o d 4 ) p\equiv 1\,(mod\, 4) . We set K = Q ( m , d ) K = Q({\sqrt {m}},{\sqrt {d}}) , where m = 2 m=2 or m = p m=p . In this paper, we determine the rank of the 2 2 -class group of K K . Résumé. Soit K = Q ( m , d ) K = Q({\sqrt {m}},{\sqrt {d}}) , un corps biquadratique où m = 2 m=2 ou bien un premier p ≡ 1 ( m o d 4 ) p\equiv 1\,(mod\,4) et d d étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du 2 2 -groupe de classes de K K .
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